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Learned optimization algorithms are promising approaches to inverse problems by leveraging advanced numerical optimization schemes and deep neural network techniques in machine learning. In this paper, we propose a novel deep neural network architecture imitating an extra proximal gradient algorithm to solve a general class of inverse problems with a focus on applications in image reconstruction. The proposed network features learned regularization that incorporates adaptive sparsification mappings, robust shrinkage selections, and nonlocal operators to improve solution quality. Numerical results demonstrate the improved efficiency and accuracy of the proposed network over several state-of-the-art methods on a variety of test problems. 

This work aims at developing a generalizable Magnetic Resonance Imaging (MRI) reconstruction method in the meta-learning framework. Specifically, we develop a deep reconstruction network induced by a learnable optimization algorithm (LOA) to solve the nonconvex nonsmooth variational model of MRI image reconstruction. In this model, the nonconvex nonsmooth regularization term is parameterized as a structured deep network where the network parameters can be learned from data. We partition these network parameters into two parts: a task-invariant part for the common feature encoder component of the regularization, and a task-specific part to account for the variations in the heterogeneous training and testing data. We train the regularization parameters in a bilevel optimization framework which significantly improves the robustness of the training process and the generalization ability of the network. We conduct a series of numerical experiments using heterogeneous MRI data sets with various undersampling patterns, ratios, and acquisition settings. The experimental results show that our network yields greatly improved reconstruction quality over existing methods and can generalize well to new reconstruction problems whose undersampling patterns/trajectories are not present during training. 

We develop a unified level-bundle method, called accelerated constrained level-bundle (ACLB) algorithm, for solving constrained convex optimization problems. where the objective and constraint functions can be nonsmooth, weakly smooth, and/or smooth. ACLB employs Nesterov’s accelerated gradient technique, and hence retains the iteration complexity as that of existing bundle-type methods if the objective or one of the constraint functions is nonsmooth. More importantly, ACLB can significantly reduce iteration complexity when the objective and all constraints are (weakly) smooth. In addition, if the objective contains a nonsmooth component which can be written as a specific form of maximum, we show that the iteration complexity of this component can be much lower than that for general nonsmooth objective function. Numerical results demonstrate the effectiveness of the proposed algorithm.